Set-Valued Var. AnalDOI 10.1007/s11228-015-0345-4

Partial Differential Equation for Evolutionof Star-Shaped Reachability Domainsof Differential Inclusions

Stanislav Mazurenko1

Received: 9 December 2014 / Accepted: 31 August 2015© Springer Science+Business Media Dordrecht 2015

Abstract The problem of reachability for differential inclusions is an active topic in therecent control theory. Its solution provides an insight into the dynamics of an investigatedsystem and also enables one to design synthesizing control strategies under a given opti-mality criterion. The primary results on reachability were mostly applicable to convex sets,whose dynamics is described through that of their support functions. Those results werefurther extended to the viability problem and some types of nonlinear systems. However,non-convex sets can arise even in simple bilinear systems. Hence, the issue of noncon-vexity in reachability problems requires a more detailed investigation. The present articlefollows an alternative approach for this cause. It deals with star-shaped reachability sets,describing the evolution of these sets in terms of radial (Minkowski gauge) functions. Thederived partial differential equation is then modified to cope with additional state constraintsdue to on-line measurement observations. Finally, the last section is on designing optimalclosed-loop control strategies using radial functions.

Keywords Reachability sets · Differential inclusion · Star-shaped sets · Radial (gauge)function · Viability · Optimal control synthesis

Mathematics Subject Classification (2010) MSC 93B03 · MSC 49K15

1 Introduction

The present problems of control systems theory and its numerous applications includethose of reachability, estimation and feedback control, as well as those for systems sub-jected to uncertainty and incomplete information. The typical mathematical models for

� Stanislav Mazurenkostasmazurenko@mail.ru

1 Faculty of Science, RECETOX, Loschmidt Laboratories,Masaryk University, Brno, Czech Republic

S. Mazurenko

system dynamics of such systems involve set-valued analysis, particularly the notion ofset-valued trajectory tubes whose evolution in time is described by differential inclusions,Hamilton-Jacobi-Bellman techniques or funnel equations [1–9]. Reachability is well stud-ied for convex sets, and the obtained results were significantly extended to the viabilityproblem and some types of nonlinear systems [2, 5, 10, 11].

Moreover, some of the previous results are applicable to non-convex sets that still havesome convex-like properties. In particular, the set-valued equation for reachability domainsin [5] remains exact for star-shaped sets, i.e. sets with some particular point (the center) thatcan be connected to all other points of the set by a straight-line segment belonging to the set.However, support functions used for describing convex sets can no longer be applied in thiscase as they will reduce the sets to their convex hulls. An alternative method was suggestedin [12], where the author showed that under some assumptions a continuously differentiableright-hand side of a differential equation

dx

dt= f (t, x; a),

with a from a convex compact set A results in the evolution of the initial set that canbe described in terms of radial (Minkowski gauge) functions. The author derived a partialdifferential equation for this radial function through the use of Lagrange multipliers:

∂+r(l, t)

∂t= max

a∈A

⟨−∂ ln r(l, t)

∂l, f (t, r(l, t)l; a)

⟩

and demonstrated how this equation changes in the presence of state constraints.Nonetheless, the derived partial differential equation does not have any derivatives of

f, hence a continuous differentiability of the right-hand side seems redundant. Indeed, thepresent article follows [12] describing the evolution of reachability sets in terms of theradial functions as opposed to support functions in the convex case but replaces the con-dition of differentiability with Lipschitz continuity at the expense of Lagrange multipliers.It further extends the result onto differential inclusions providing a new derivation of thepartial differential equation for a radial function using fixed-point and implicit function the-orems. Similar to [12], in this article we show how the resulting partial differential equationis affected by introduction of state constraints. In addition to that, we now also comparethe derived method for finding reachability sets with the existing one that uses Hamilton-Jacobi-Bellman equation for a specific value function. Finally, in the last section we applythe results of the previous sections to optimal control problems and provide an algorithmfor designing optimal control synthesis.

2 Notations

The following notations will be used throughout this paper:

– Rn - the n−dimensional Euclidean space;– 〈x, y〉 - the inner product of x, y ∈ Rn;– ‖x‖ = 〈x, x〉1/2;– int A - the interior of A ⊆ Rn;– ∂A - the boundary of A ⊆ Rn, i.e. ∂A = A \ int A;– comp Rn - the variety of all compact subsets of Rn;– conv Rn - the variety of all compact convex subsets of Rn;

PDE for Evolution of Star-Shaped Reachability Domains

– St (Rn) - the variety of all star-shaped compact subsets of Rn, i.e. Z ∈ St (Rn) ⇐⇒Z ∈ comp Rn and ∀λ ∈ [0, 1] ⇒ λZ ⊆ Z;

– B0,1 - the unit ball in Rn, i.e. B0,1 = {x ∈ Rn : ‖x‖ ≤ 1};– h(X, Y ) - the Hausdorff distance for X, Y ⊆ Rn :

h(X, Y ) = inf{ε ≥ 0 : X ⊆ Y + εB0,1, Y ⊆ X + εB0,1

};

– ρ(l|X) - the support function of X ⊆ Rn at point l ∈ Rn :ρ(l|X) = sup{〈l, x〉| x ∈ X};

– r(l|X) - radial function of X ⊆ Rn at point l ∈ Rn :r(l|X) = sup{λ ∈ R1 : λl ∈ X}.

3 Differential Inclusions and Uncertain Systems

This article deals with dynamical systems that can be described as a set of differentialequations with some unknown parameters. Consider the following differential equation:

dx

dt= f (t, x, v)

where f : R1 × Rn × Rp → Rn is a measurable function, x is a state vector and v is aninput parameter, which may stand for an external control or disturbance. The values of v areassumed to be unknown but bounded: v ∈ V (t, x). This problem can thus be rewritten as adifferential inclusion

dx

dt∈ F(t, x) =

⋃v∈V (t,x)

f (t, x, v).

We are interested in studying the set of the solutions x[t] = x(t, t0, x0) to this differentialinclusion that originate from a compact set X0 at time t0 :

x[t0] = x0 ∈ X0.

Let X[t] denote the set of points x[t] for various starting point x0 ∈ X0. If X[t] is convexat any instant t ∈ T , it is known [5, 10] that the evolution of this set can be expressed in termsof its support function ρ(l, t) = ρ(l|X[t]) that satisfies the following partial differentialequation:

∂+ρ(l, t)

∂t= max

x∈X[t]:〈l,x〉=ρ(l,t)ρ(l|F(t, x)), (1)

where ∂+ρ(l,t)∂t

stands for the right directional derivative in t .However, if the reachability set of a system is not convex, the above equation defines

evolution of the convex hull of reachability domain. In order to demonstrate that the hullcan provide poor approximation of the exact set consider the following trivial example.

Example 1 ⎧⎨⎩

x = 0,y = ux, u ∈ [−1, 1],x(0) ∈ [−1, 1], y(0) = 0.

Here the convex initial set X0 = [−1, 1] × {0} results in the non-convex reachability setfor any time t > 0. Indeed, take t = 1 :

X[t = 1] = {(x, y) : |x| ≤ 1, |y| ≤ |x|}.

S. Mazurenko

This set is non-convex, e.g. (±1, 1) ∈ X[t = 1] but (0, 1) /∈ X[t = 1]. Moreover, thefollowing set comprises the convex hull but does not belong to the exact set: {(x, y) : |x| ≤1, 1 − |x| < |y| ≤ 1}\{(±1,±1)}.

There have been attempts to derive equations similar to (1) [2], however due to computa-tional complexity and sophisticated structure of sets, alternative approximate solutions areusually used instead [7, 13, 15].

The purpose of this work is to derive a differential equation of evolution of reachabilitydomains in terms of a function that possesses some properties of the support function butcan grasp the non-convexity of the reachability set. We will use radial functions (the inverseof Minkowski or gauge functions [14]) in the following form:

r(l|Z) = sup{λ ∈ R1 : λl ∈ Z}.

The radial function is an efficient method of describing non-convex star-shaped sets asthere are quite a few useful properties.

Proposition 1 For any compact sets Z1 ⊆ Z2 and any non-zero l ∈ Rn it follows that

r(l|Z1) ≤ r(l|Z2).

Proposition 2 For any Z ∈ St (Rn) it follows that

Z = {l ∈ Rn \ {0} : r(l|Z) ≥ 1}⋃{0}.

Proposition 3 For any Zi ∈ St (Rn), i = 1, . . . , m and any non-zero l ∈ Rn it follows that

r

⎛⎝l

∣∣∣∣∣∣⋃

i=1,...,m

Zi

⎞⎠ = max

i=1,...,mr(l|Zi).

Proposition 4 For any Zi ∈ St (Rn), i = 1, . . . , m and any non-zero l ∈ Rn it follows that

r

⎛⎝l

∣∣∣∣∣∣⋂

i=1,...,m

Zi

⎞⎠ = min

i=1,...,mr(l|Zi).

Proposition 5 For any Z ∈ St (Rn) and λ > 0 it follows that

r(λl|Z) = r

(l

∣∣∣∣1λZ

)= 1

λr(l|Z).

The above statements are quite straightforward and their proofs are left as an exercise tothe reader.

Proposition 6 For any Z ∈ St (Rn) such that {0} ∈ int Z and r(l|Z) is continuous in l forany non-zero l ∈ Rn it follows that

∂Z = {l ∈ Rn : r(l|Z) = 1}.

PDE for Evolution of Star-Shaped Reachability Domains

Proof Let Q = {l ∈ Rn : r(l|Z) = 1}. On the one hand, it immediately follows thatQ ⊆ Z. But

∀z ∈ int Z ∃ε > 0 : (1 + ε)z ∈ Z ⇒ r((1 + ε)z|Z) = 1

1 + εr(z|Z) ≥ 1 ⇒

⇒ r(z|Z) > 1 ⇒ Q⋂

int Z = ∅.

Hence, Q ⊆ ∂Z.On the other hand, for any z ∈ ∂Z there exists zn → z, zn /∈ Z, consequently, due to

continuity of r

r(zn|Z) < 1 ⇒ r(z|Z) = limn→+∞ r(zn|Z) ≤ 1.

But z ∈ Z ⇒ r(z|Z) ≥ 1 so r(z|Z) = 1 ⇒ ∂Z ⊆ Q.

Corollary 1 Due to homogeneity of r the boundary of Z can be parameterized as follows

∂Z =⋃

k∈∂B0,1

{r(k|Z)k} .

Proposition 7 For any Z ∈ St (Rn) such that {0} ∈ int Z it follows that r(l|Z) isdifferentiable in the direction of l for any non-zero l ∈ Rn and its directional derivative⟨

∂r(l|Z)

∂l, l

⟩= −r(l|Z).

Proof By definition⟨∂r(l|Z)

∂l, l

⟩= lim

δ→0

r(l + δl|Z) − r(l|Z)

δ= lim

δ→0

11+δ

r(l|Z) − r(l|Z)

δ= −r(l|Z).

Proposition 8 If the set Z is not star-shaped, the following set is the star-shaped hull of Z :SH(Z) = {l ∈ Rn \ {0} : r(l|Z) ≥ 1

}⋃{0},that is SH(Z) ∈ St (Rn), Z ∈ SH(Z) and for any Z ∈ St (Rn) : Z ⊆ Z it follows thatSH(Z) ⊆ Z.

Proof Since r(l|Z) is homogeneous, for any positive λ ≤ 1 it follows that λSH(Z) ⊆SH(Z). Moreover, for any l ∈ Z \ {0} : r(l|Z) ≥ 1 by definition of r, thus Z ⊆ SH(Z).Finally, for any Z ∈ St (Rn) : Z ⊆ Z it follows that

r(l|Z) ≥ r(l|Z)

andZ ={l ∈ Rn \ {0} : r(l|Z) ≥ 1

}⋃{0}.

But for any l ∈ SH(Z) ⇒ r(l|Z) ≥ 1 ⇒ r(l|Z) ≥ 1 ⇒ l ∈ Z ⇒ SH(Z) ⊆ Z.

4 Evolution Equation

Consider the following differential inclusion:

x ∈ F(t, x), x(t0) = x0 ∈ X0 ∈ comp Rn, t ∈ T = [t0, t1], (2)

S. Mazurenko

where F is a continuous multivalued map T × Rn → conv Rn that satisfies the Lipschitzcondition with constant L > 0 :

∀x, y ∈ Rn ⇒ h(F (t, x), F (t, y)) ≤ L‖x − y‖.We are interested in a Caratheodory-type solutions to (2), namely an absolutely con-

tinuous functions x[t] = x(t, t0, x0) that satisfy (2) for almost every t ∈ [t0, t1] andx[t0] = x0 ∈ X0. We will also assume that for every point x0 ∈ X0 the solution x[t] existsin the whole time interval [t0, t1].

Let X(·, t0, X0) be the set of all the solutions and X[t] be its cross-section at time t . Itcan be shown [16] that under the above conditions X[t] = X(t, t0, X0) is a compact-valuedmap (X : T × T × comp Rn → comp Rn) that satisfies a semigroup property:

∀τ : t0 ≤ τ ≤ t ≤ t1 ⇒ X(t, τ, X(τ, t0, X0)) = X(t, t0, X0).

Theorem 1 [1] Under the above conditions for (2), the reachability set X[t] is the onlycompact-set solution to the following equation:

limσ→+0

σ−1h

⎛⎝X[t + σ ],

⋃x∈X[t]

{x + σF(t, x)}⎞⎠ = 0, t ∈ T , X[t0] = X0. (3)

Due to computational limitations to (3) we would like to obtain the result in terms offunctions rather than set-valued operators. In the case of convex sets X[t] the partial dif-ferential equation in terms of support functions is available [10] as was outlined above.However, when the sets are not convex, the support function may be replaced by a radialfunction, which will be demonstrated further.

Consider the following assumption:

Property 1 1. For every t ∈ T graphtF = {(x, y) ∈ R2n : y ∈ F(t, x)} ∈ St (R2n).2. The set X0 ∈ St (Rn).3. There exists r− > 0 : ∀t ∈ T ⇒ r−B0,1 ⊆ X[t].

The first two conditions result in X[t] ∈ St (Rn) for any time t ∈ T . The third conditionimplies that the centre of the star (the origin in this paper) never appears on the boundaryof the set, hence, the radial function r(l, t) = r(l|X[t]) is always greater or equal to somepositive constant ε, which is a technical condition required because in general the radialfunction is ill-defined for sets that do not comprise the centre of the star. For example, thiscondition is met if ∃ε > 0 : ∀t ∈ T ⇒ εB0,1 ⊆ F(t, 0). The case of the centre differentfrom the origin will be touched upon later on, but the idea is that the result of this paper canbe modified to account for a moving centre described by a pre-defined continuously differ-entiable function. However, the derivation of this function for a given differential inclusionis beyond the scope of this study.

Theorem 2 Suppose that for (2) Property 1 holds. Then for any non-zero direction l ∈ Rn,

for which the radial function of the reachability set r(l, t) = r(l|X[t]) is continuouslydifferentiable in l, there exits ∂+r(l, t)/∂t that satisfies the following equation

∂+r(l, t)

∂t= 1

r(l, t)ρ

(−∂r(l, t)

∂l

∣∣∣∣F(t, r(l, t)l)

). (4)

PDE for Evolution of Star-Shaped Reachability Domains

Proof The proof of the result will be implemented in several steps. But before we start wewill need the following three technical lemmas.

Lemma 1 There exists σ > 0 such that for any function g(σ ) : R1 → Rn positive σ <

σ , λ ∈ R1, l ∈ Rn there exists a fixed point y ∈ Rn :y ∈ F(t, λl − σy + g(σ )). (5)

Moreover, y = y(λ) can be made a continuous function of λ that passes through anygiven fixed point of (5).

Indeed, due to the Lipschitz condition on F it follows that

∀y1, y2 ∈ Rn h(F (t, λl − σy2 + g(σ )), F (t, λl − σy1 + g(σ )) ≤ Lσ‖y2 − y1‖.Hence, for σ ≤ σ = 1/2L the mapping y → F(t, λl − σy + g(σ )) is a multi-valued

contraction mapping with Lipschitz constant no greater than 1/2, consequently it has a fixedpoint [17]. Moreover, since F ∈ conv Rn, is continuous in (λ, y) and its Lipschitz constantis independent of λ, we can choose fixed points that comprise a continuous function of λ,

which passes through any given fixed point of (5) [18, 19].

Lemma 2 Suppose X ∈ comp Rn. Then for any function g(σ ) : R1 → R1 and non-zerodirection l ∈ Rn there exists σ > 0 such that for any positive σ < σ

r = r

(l

∣∣∣∣∣⋃x∈X

{x + σF(t, x) + g(σ )B0,1})

> r

(l

∣∣∣∣∣⋃

x∈int X{x + σF(t, x) + g(σ )B0,1}

).

In other words, this lemma says that the following maximization problem

max{λ ∈ R1 : λl = x + σy + g(σ )z, x ∈ X, y ∈ F(t, x), z ∈ B0,1}attains its maximum on the boundary of X (notice that the second argument of r on the lefthand side of the above inequality is compact). It is quite obvious that the optimal y and z

must be at the boundary of the respective sets, otherwise we can shift them in the directionl to increase λ. However, the same result for x is not that obvious as for different values ofx ∈ X we have in general different possible values of y ∈ F(t, x).

Nonetheless, for Lipschitz mapping F the optimal x ∈ ∂X for small σ . Indeed, supposethis does not hold, i.e. there exists σn → +0, xn ∈ int X, yn ∈ F(t, xn), zn ∈ B0,1 such thatfor the maximal λn

xn = λnl − σnyn − g(σn)zn ∈ int X.

According to the previous lemma, there exists a continuous function

yn(δ) ∈ F(t, (λn + δ)l − σnyn(δ) − g(σn)zn), yn(0) = yn

for all the large numbers of n such that σn < σ . Then consider xn = (λn + δ)l − σnyn(δ) −g(σn)zn :

‖xn − xn‖ = ‖δl − σn(yn(δ) − yn)‖ ≤ δ‖l‖ + σn‖yn(δ) − yn‖ → 0 for δ → 0.

Since xn ∈ int X, for a small positive δ it follows that xn ∈ X. Hence, there is xn ∈ X,

yn ∈ F(t, xn), and zn = zn ∈ B0,1, for which λn can be increased by δ which contradictsto the fact that λn is maximal.

S. Mazurenko

Lemma 3 Suppose X ∈ comp Rn, {0} ∈ intX, r(l|X) is continuous in l, and for a givenfunction g(σ, x) : Rn+1 → Rn there exists C such that for all x ∈ X and small valuesσ ‖g(σ, x)‖ ≤ C. For a non-zero direction l ∈ Rn consider the following maximizationproblem:

λ → maxx∈X

, s.t. : λl = x + σg(σ, x).

Suppose that there exists optimal x∗(σ ) ∈ ∂X for any small value of σ . Thenlimσ→0

x∗(σ ) = r(l|X)l.

Since r(l|X) is continuous it follows that r(x∗|X) = 1, we can represent each x ∈ ∂X

as x = r(k|X)k for some non-zero k ∈ B0,1 (Corollary 1). Consider the orthogonal to l

component of optimal k∗, i.e. k∗ − 〈l,k∗〉‖l‖2 l. Since λl = x + σg(σ, x) it follows that

0 =⟨λl, k∗ − 〈l, k∗〉

‖l‖2 l

⟩= r(k∗|X)

∥∥∥∥k∗ − 〈l, k∗〉‖l‖2 l

∥∥∥∥2

+ σ

⟨g(σ, x∗), k∗ − 〈l, k∗〉

‖l‖2 l

⟩.

The last item goes to zero as both g and k∗ are bounded. But for any limit point k of

k∗(σ ) the radial function r(k|X) > 0 as {0} ∈ intX, so∥∥∥k − 〈l,k〉

‖l‖2 l

∥∥∥ = 0 ⇒ k = αl ⇒r(k|X)k = r(l|X)l.

Thus, limσ→0

x∗(σ ) = r(l|X)l.

Now we can proceed to the proof of the theorem. We will use Theorem 1:

limσ→+0

σ−1h

⎛⎝X[t + σ ],

⋃x∈X[t]

{x + σF(t, x)}⎞⎠ = 0,

which is equivalent to the following two inclusions:

X[t + σ ] ⊆⋃

x∈X[t]{x + σF(t, x)} + σoσ (1)B0,1, (6)

⋃x∈X[t]

{x + σF(t, x)} ⊆ X[t + σ ] + σoσ (1)B0,1, (7)

where limσ→+0

oσ (1) = 0.

Now consider l ∈ Rn, for which the radial function r(l, t) is continuously differentiablein l. In terms of radial functions, inclusion (6) is equivalent to

r(l, t + σ) ≤ r

⎛⎝l

∣∣∣∣∣∣⋃

x∈X[t]{x + σF(t, x)} + σoσ (1)B0,1

⎞⎠ .

By definition of the radial function, the right-hand side of the above inequality isequivalent to the following maximization problem:

max{λ ∈ R1 : λl = x + σy + σoσ (1)z, x ∈ X[t], y ∈ F(t, x), z ∈ B0,1}Let x∗(σ ) denote the optimal x and λ∗(σ ) denote the maximal value of λ. For σ = 0 itimmediately follows that the optimal x∗ = λ∗l = r(l, t)l. Since F(t, x) is Lipschitz andX[t] is compact, the set

⋃x∈X[t]

F(t, x) is bounded. As a result, for small values of σ the

PDE for Evolution of Star-Shaped Reachability Domains

optimal x∗ is at the boundary of X[t] according to Lemma 2, and consequently,x∗ → r(l, t)l according to Lemma 3. Let us fix an arbitrary ε > 0 and the respectivevicinity of r(l, t)l + εB0,1. Consider Fε :

Fε =⋃{

F(t, x)

∣∣∣x ∈ ∂X[t]⋂{

r(l, t)l + εB0,1}}

.

Then for small values of σ it follows that λ∗(σ ) ≤ λε(σ ), where

λε(σ ) = max{λ ∈ R1 : λl = x + σy + σoσ (1)z

∣∣∣x ∈ ∂X[t]

⋂{r(l, t)l + εB0,1

}, y ∈ Fε, z ∈ B0,1

}.

We will now separate σ to be able to cope with possible non-smoothness of oσ (1). Letλε(σ ) = λε(σ, σ ), where

λε(σ1, σ2) = maxy∈Fε,z∈B0,1

λ(σ1, σ2; y, z) =

= maxy∈Fε,z∈B0,1

max{λ ∈ R1 : λl = x + σ1y + σ1oσ2(1)z

∣∣∣ x ∈ ∂X[t]⋂{

r(l, t)l+εB0,1}}

.

Then the optimal x∗(σ1, σ2; y, z) = λl − σ1y − σ1oσ2(1)z. Since it must belong to theboundary of X[t] (otherwise, we could increase λ moving x in the l direction)

r(λl − σ1y − σ1oσ2(1)z, t) = 1, (8)

and we can apply Lemma 3 to the inner maximum:

limσ→+0

x∗(σ, σ ; y, z) = r(l, t)l.

But r is continuously differentiable in (λ, σ1) and (for simplicity of formulas we will omitt in r(l, t))

∂r

∂λ(x∗) =

⟨∂r

∂l(x∗), l⟩

= 1

r(l, t)

(⟨∂r

∂l(x∗), x∗

⟩+⟨∂r

∂l(x∗), r(l, t)l − x∗

⟩)=

= − 1

r(l, t)+ 1

r(l, t)

⟨∂r

∂l(x∗), r(l, t)l − x∗

⟩.

For small values of ε the second item goes to zero and r(l, t) is limited from above byr−. Hence, we can apply the implicit function theorem to (8) with respect to λ and obtainthe following result:

∂λ(σ1, σ2; y, z)

∂σ1=⟨∂r

∂l(x∗), l⟩−1 ⟨

∂r

∂l(x∗), y + oσ2(1)z

⟩.

Moreover, since λ(0, σ2; y, z) = r(l, t) we can apply Taylor’s theorem with Peano formof the reminder:

λ(σ1, σ2; y, z) = r(l, t) +⟨∂r

∂l(x∗), l⟩−1 ⟨

∂r

∂l(x∗), y + oσ2(1)z

⟩σ1 + σ1oσ1(1) ⇒

λ(σ, σ ; y, z) = r(l, t) +⟨∂r

∂l(x∗), l⟩−1 ⟨

∂r

∂l(x∗), y⟩σ+

+(⟨

∂r

∂l(x∗), l⟩−1 ⟨

∂r

∂l(x∗), z⟩+ 1

)σoσ (1).

S. Mazurenko

Hence, from (6) it follows that

r(l, t + σ) ≤ r(l, t) + σ maxy∈Fε,z∈B0,1

(⟨∂r

∂l(x∗, t), l

⟩−1 ⟨∂r

∂l(x∗, t), y

⟩+

+(⟨

∂r

∂l(x∗, t), l

⟩−1 ⟨∂r

∂l(x∗, t), z

⟩+ 1

)oσ (1)

)⇒ (9)

r(l, t + σ) − r(l, t)

σ≤ max

y∈Fε,z∈B0,1

(⟨∂r

∂l(x∗, t), l

⟩−1 ⟨∂r

∂l(x∗, t), y

⟩+

+(⟨

∂r

∂l(x∗, t), l

⟩−1 ⟨∂r

∂l(x∗, t), z

⟩+ 1

)oσ (1)

).

Now we take a limit of σ → +0 and taking into account that x∗ → r(l, t)l we arrive atthe following conclusion:

limσ→+0

r(l, t + σ) − r(l, t)

σ≤ max

y∈Fε

⟨∂r

∂l(r(l, t)l, t), l

⟩−1 ⟨∂r

∂l(r(l, t)l, t), y

⟩,

or using the homogeneity of the radial function:

limσ→+0

r(l, t + σ) − r(l, t)

σ≤ max

y∈Fε

⟨− 1

r(l, t)

∂r

∂l(l, t), y

⟩.

Since the above inequality holds for any positive ε we can take infimum:

limσ→+0

r(l, t + σ) − r(l, t)

σ≤ inf

ε>0maxy∈Fε

⟨− 1

r(l, t)

∂r

∂l(l, t), y

⟩=

= maxy∈F(t,r(l,t)l)

⟨− 1

r(l, t)

∂r

∂l(l, t), y

⟩= 1

r(l, t)ρ

(−∂r(l, t)

∂l

∣∣∣∣F(t, r(l, t)l)

).

Now consider inclusion (7), which can be rewritten as follows:

r

⎛⎝l

∣∣∣∣∣∣⋃

x∈X[t]{x + σF(t, x)}

⎞⎠ ≤ r(l|X[t + σ ] + σoσ (1)B0,1). (10)

As far as the right-hand side of this inequality is concerned,

r(l|X[t +σ ]+σoσ (1)B0,1) = max{λ ∈ R1 : λl = x +σoσ (1)z|x ∈ X[t +σ ], z ∈ B0,1} =

= maxz∈B0,1

λ+(σ1, σ2; z)

∣∣∣∣σ1=σ2=σ

,

whereλ+(σ1, σ2; z) = max{λ ∈ R1 : λl = x + σ1oσ2(1)z|x ∈ X[t + σ2]}.

Again it is easy to see that the optimal x+(σ1, σ2; z)must be at the boundary ofX[t+σ2],and applying Lemma 3 we get lim

σ1→+0x+(σ1, σ2; z) = r(l, t +σ2)l. Hence, λ+ can be found

as an implicit function from the following equation

r(λl − σ1oσ2(1)z, t + σ2) = 1.

If we follow the similar steps as above (Implicit function theorem and Taylor’s theoremwith Peano form of the reminder) we will obtain the following decomposition:

PDE for Evolution of Star-Shaped Reachability Domains

λ(σ1, σ2; y, z) = r(l, t + σ2)+

+⟨∂r

∂l(x+, t + σ2), l

⟩−1 ⟨∂r

∂l(x+, t + σ2), z

⟩σ1oσ2(1) + σ1oσ1(1) ⇒

r(l|X[t + σ ] + σoσ (1)B0,1) = r(l, t + σ)+

+(maxz∈B0,1

⟨∂r

∂l(x+, t + σ), l

⟩−1 ⟨∂r

∂l(x+, t + σ), z

⟩+ 1

)σoσ (1) =

= r(l, t + σ) + σoσ (1).

The last equality follows from the fact limσ→+0

x+(σ, σ ; z) = r(l, t)l, in the vicinity of

which function r(l, t) is continuously differentiable (consequently, its partial derivative isbounded and Proposition 7 holds). Since this result will be used for other sections we willformulate it as a separate lemma:

Lemma 4 If Z(t) ∈ St (Rn), and function r(l|Z(t)) ≥ rz > 0 and is continuously dif-ferentiable at some non-zero direction l as a function of (l, t). Then the following equalityholds:

r(l|Z(t + σ) + σoσ (1)B0,1) = r(l|Z(t + σ)) + σoσ (1).

Now consider the left-hand side of inequality (10):

r

⎛⎝l

∣∣∣∣∣∣⋃

x∈X[t]{x + σF(t, x)}

⎞⎠ = r

⎛⎝l

∣∣∣∣∣∣⋃

x∈∂X[t]{x + σF(t, x)}

⎞⎠

here we again applied Lemma 2 with g ≡ 0. The latter radial function is equivalent to asolution to the following optimization problem:

λ → maxx∈∂X[t],y∈F(t,x)

, s.t. : λl = x + σy,

which is now equivalent to

λ → maxy∈Rn

, s.t. : r(λl − σy, t)=1, y ∈ F(t, λl − σy). (11)

According to Lemma 3 the optimal x− = λl − σy satisfies limσ→+0

x−(σ ) = r(l, t)l.

Again for any y the equation r(λl − σy, t) = 1 defines an implicit function λ−(σ, y) thatafter applying Taylor’s theorem with Peano form of the remain can be rewritten as

λ−(σ, y) = r(l, t) +⟨∂r

∂l(x−, t), l

⟩−1 ⟨∂r

∂l(x−, t), y

⟩σ + σoσ (1).

In this case the maximal value of λ−(σ ) in (11) is

λ−(σ ) = maxy∈F [σ,y] λ

−(σ, y),

where F [σ, y] = F(t, λ−(σ, y)l − σy). Since limσ→+0

x−(σ ) = r(l, t)l, and r(l, t) is contin-

uously differentiable (consequently, its partial derivative is bounded), there exists a positiveconstant K : ∥∥∥∥∥

⟨∂r

∂l(x−, t), l

⟩−1∂r

∂l(x−, t)

∥∥∥∥∥ ≤ K

for all small σ . Hence, for any y1, y2 ∈ Rn

S. Mazurenko

h(F [σ, y1], F [σ, y2]) ≤ L(‖(λ−(σ, y1) − λ−(σ, y2))l‖ + σ‖y1 − y2‖) ≤≤ L(Kσ‖y1 − y2‖‖l‖ + σ‖y1 − y2‖) = σL(K‖l‖ + 1)‖y1 − y2‖.

For small σ it follows that σL(K‖l‖+1) < 1/2 and F [σ, y] is a multi-valued contractionmapping with Lipschitz constant 1/2, consequently it has a fixed point [17]. Moreover,since F [σ, y] is continuous in (σ, y) and its Lipschitz constant is independent of σ, we canchoose a continuous function y(σ ) : y(σ ) ∈ F [σ, y(σ )] for any starting point y(0) = y0 ∈F(t, r(l, t)l) [18, 19]. Consequently,

r

⎛⎝l

∣∣∣∣∣∣⋃

x∈X[t]{x + σF(t, x)}

⎞⎠ = r(l, t) + max

y∈F [σ,y]

⟨∂r

∂l(x−, t), l

⟩−1 ⟨∂r

∂l(x−, t), y

⟩σ+

+ σoσ (1) ≥ r(l, t) +⟨∂r

∂l(x−, t), l

⟩−1 ⟨∂r

∂l(x−, t), y(σ )

⟩σ + σoσ (1). (12)

Therefore, inequality (10) can be rewritten as

r(l, t) +⟨∂r

∂l(x−, t), l

⟩−1 ⟨∂r

∂l(x−, t), y(σ )

⟩σ + σoσ (1) ≤ r(l, t + σ) + σoσ (1),

i.e.

r(l, t + σ) − r(l, t)

σ≥⟨∂r

∂l(x−, t), l

⟩−1 ⟨∂r

∂l(x−, t), y(σ )

⟩+ oσ (1).

If now σ goes to zero, we arrive at the following inequality

limσ→+0

r(l, t + σ) − r(l, t)

σ≥⟨∂r

∂l(r(l, t)l, t), l

⟩−1 ⟨∂r

∂l(r(l, t), t), y0

⟩,

or using the homogeneity of the radial function:

limσ→+0

r(l, t + σ) − r(l, t)

σ≥⟨− 1

r(l, t)

∂r

∂l(l, t), y0

⟩.

Since the above inequality holds for any y0 ∈ F(t, r(l, t)l), we can take a maximum:

limσ→+0

r(l, t + σ) − r(l, t)

σ≥ max

y0∈F(t,r(l,t)l)

⟨− 1

r(l, t)

∂r

∂l(l, t), y0

⟩=

= 1

r(l, t)ρ

(−∂r(l, t)

∂l

∣∣∣∣F(t, r(l, t)l)

).

Combining the two inequalities we obtain the result of the theorem.

Theorem 3 Suppose that for (2) Property 1 holds. Suppose r(l, t) is homogeneous in l,

continuously differentiable in (l, t), and satisfies

∂+r(l, t)

∂t= 1

r(l, t)ρ

(−∂r(l, t)

∂l

∣∣∣∣F(t, r(l, t)l)

), r(l, t0) = r(l|X0)

for any non-zero direction l ∈ Rn. Let Z(t) be the set defined by the function r(l, t), i.e.

Z(t) = {l ∈ Rn \ {0} : r(l, t) ≥ 1}⋃{0}.

Then Z(t) ∈ St (Rn) and satisfies

limσ→+0

σ−1h

⎛⎝Z(t + σ),

⋃x∈Z(t)

{x + σF(t, x)}⎞⎠ = 0, t ∈ T , Z(t0) = X0.

PDE for Evolution of Star-Shaped Reachability Domains

Notice that according to Theorem 1 it then will follow that Z(t) is the reachability setfor (2).

Proof The fact that Z(t) ∈ St (Rn) follows from continuity and homogeneity of r(l, t) in l.Moreover, due to homogeneity the differential equation in differences may be rewritten as

r(l, t + σ) = r(l, t) + σ maxy∈F(r(l,t)l,t)

{⟨∂r(l, t)

∂l, l

⟩−1 ⟨∂r(l, t)

∂l, y

⟩}+ o(σ ). (13)

The required equation onZ(t) is tantamount to (6) and (7), which in turn can be rewrittenas (9) and (12) respectively since r(l, t) = r(l|Z(t)) is continuously differentiable in l andconsequently all the required lemmas hold. In order to prove that from (13) it follows that(9) and (12) we need to show that∣∣∣∣∣σ max

y∈F(r(l,t)l,t)

{⟨∂r(l, t)

∂l, l

⟩−1 ⟨∂r(l, t)

∂l, y

⟩}−

−σ maxy∈F [σ,y]

{⟨∂r(x−, t)

∂l, l

⟩−1 ⟨∂r(x−, t)

∂l, y

⟩}∣∣∣∣∣ = o(σ )

and ∣∣∣∣∣σ maxy∈F(r(l,t)l,t)

{⟨∂r(l, t)

∂l, l

⟩−1 ⟨∂r(l, t)

∂l, y

⟩}−

−σ maxy∈Fε

{⟨∂r(x∗, t)

∂l, l

⟩−1 ⟨∂r(x∗, t)

∂l, y

⟩}∣∣∣∣∣ = o(σ )

Sincelim

σ→+0x∗ = lim

σ→+0x− = r(l, t)l,

we will only have show that∣∣∣∣ maxy∈F(r(l,t)l,t)

⟨∂r(l, t)

∂l, y

⟩− max

y∈Fσ

⟨∂r(xσ , t)

∂l, y

⟩∣∣∣∣ = o(1),

or, in other words,

limσ→+0

maxy∈Fσ

⟨∂r(xσ , t)

∂l, y

⟩= max

y∈F(r(l,t)l,t)

⟨∂r(l, t)

∂l, y

⟩

where xσ and Fσ are x∗, Fε(σ) or x−, F [σ, y(xσ )] respectively.Indeed, consider arbitrary σn → σ and respective xn and yn ∈ Fσn . Then for any

limit point y∗ of the latter succession it follows that y∗ ∈ F(r(l, t)l, t) since Fσn isLipshitz-continuous. As a result, we can go to the limit at σ = 0 and obtain the necessaryequation.

Thus, the evolution of star-shaped reachability domains can be derived via equation (4).Now we will elaborate on the main constraints of the result. The first important requirementis the Lipschitz continuity of the right-hand side of the differential equation. The violationmight result in attainability of maximum in Lemma 2 in the interior of the set X[t], hencethe limit for vanishingly small values of σ may no longer exist and the radial function mightnot be differentiable in t .

Next requirement is continuous differentiability of the radial function r(l, t) in thevicinity of the chosen direction l. Consider the following example:

S. Mazurenko

Example 2 Let the differential inclusion be defined as

x ∈ F(t, x) =⋃

u1∈[0,a], u2∈E(b,B)

{u1x + u2} ,

where x ∈ Rn, a > 0, E(b, B) is an ellipsoid centred at b with the matrix B such that

ρ(l|E(b, B)) = 〈l, b〉 + 〈l, Bl〉 12 ≥ 0. In this case it is easy to demonstrate that Property 1

holds for the appropriate star-shaped initial set X0 and (4) will be as follows:

∂+r(l, t)

∂t= ρ

(− 1

r(l, t)

∂r(l, t)

∂l

∣∣∣∣F(t, r(l, t)l)

)=

= maxu1∈[0,a]

{⟨−∂r(l, t)

∂l, u1l

⟩}+ ρ

(− 1

r(l, t)

∂r(l, t)

∂l

∣∣∣∣ E(b, B))

)=

= ar(l, t) − 1

r(l, t)

⟨∂r(l, t)

∂l, b

⟩+ 1

r(l, t)

⟨∂r(l, t)

∂l, B

∂r(l, t)

∂l

⟩ 12

.

For the trivial case b = 0, B = 0 it immediately follows that r(l, t) = r(l, t0)ea(t−t0),

i.e. the radial function is continuously differentiable for a continuously differentiable initialradial function. However, the more general case requires numerical integration of the aboveequation and can result in continuous but not differentiable solution. Thus, in most casesthe presence of support function on the right-hand side of (4) may result in continuous onlysolution even for the given differentiable initial function. Taking into account that the (4)is proper degenerate elliptic, it might be expedient for such cases to resort to generalizedsolutions such as viscosity ones [20], which however requires additional study. An alterna-tive way to deal with the directions where differentiability vanishes is extrapolation of thesmooth segments on each step of calculation, but this seems feasible only if the number ofsuch directions is finite.

Finally, consider the requirement that the centre of the star-shaped set be at the origin.This restriction can be lifted: suppose the trajectory of the centre is defined by a continuouslydifferentiable function c(t), for instance the following assumptions hold for a given c(t) :

Property 2 1. For every t ∈ T and 0 < λ ≤ 1 it follows that

F(t, x) ⊆ 1

λF(t, λx + (1 − λ)c(t)) − 1 − λ

λc(t).

2. The set X0 − c(t0) ∈ St (Rn).3. There exists r− > 0 : ∀t ∈ T ⇒ r−B0,1 ⊆ X[t] − c(t).

Notice that Property 1 is a special case of Property 2 with c(t) ≡ 0. A relatively easyderivation results in the following changes to (4):

∂+rc(l, t)

∂t= 1

rc(l, t)ρ

(−∂rc(l, t)

∂l

∣∣∣∣ c(t) + F(t, rc(l, t)l + c(t))

),

where rc(l, t) = r(l|X[t] − c(t)). However, the exact algorithm for derivation of c(t) forany given differential inclusion with star-shaped reachability sets is yet to be developed.

We will now proceed to comparison of the result with more general approach to theproblem of derivation reachability sets through Hamilton-Jacobi-Bellman equation on thevalue function [7]. The latter method can be summarized as follows. Consider the followingdifferential system:

x = f (t, x, u), t ∈ T = [t0, t1], u ∈ U, x(t0) ∈ X0 ∈ comp(Rn),

PDE for Evolution of Star-Shaped Reachability Domains

with standard restrictions on existence, uniqueness, and extendibility of solution x[t] ontothe time interval T . We will now fix arbitrary point x1 ∈ Rn and solve the followingoptimization problem:

V (t1, x1) = {d(x[t0], X0)| x[t1] = x1} → infu∈U

,

i.e. we would like to find the minimal distance from the starting point of a trajectory x[t] toinitial set X0 given the fixed terminal point x[t1] = x1. Then it can be shown [7] that therequired reachability set X[t] can be found as a lower contour set of V :

X[t] = {x ∈ Rn : V (t, x) ≤ 0},and the value function V satisfies the following Hamilton-Jacobi-Bellman partial differen-tial equation: {

∂V (t,x)∂t

= −maxu∈U

⟨∂V (t,x)

∂x, f (t, x, u)

⟩,

V (t0, x) = d(x,X0).

Although equation (4) might seem more complex than the above Hamilton-Jacobi-Bellman equation, the following advantages are worth noticing. First, due to homogeneityof radial function r(l, t) in l, it is sufficient to solve equation (4) on a union ball l ∈ B0,1,

whereas the Hamilton-Jacobi-Bellman equation has to be solved in Rn. Moreover, it is suf-ficient to consider l ∈ ∂B0,1, consequently, with an appropriate change of coordinates ton−dimensional spherical coordinates, one can reduce the overall dimension of the problemfrom n + 1 to n. Finally, for l ∈ ∂B0,1 r(l, t) represents intuitively a ‘radius’ of the setwhereas V (t, x) has a rather complicated structure.

Consider n−dimensional spherical coordinates l = R · e(φ), where ‖e(φ)‖ = 1, φ =(φ1, . . . , φn−1), φ1 ∈ [0, 2π), φi ∈ [0, π ] for i = 2, 3, . . . , n − 1, R > 0.⎛

⎜⎜⎜⎝

∂r∂φ1

. . .∂r

∂φn−1∂r∂R

⎞⎟⎟⎟⎠ =

⎛⎜⎜⎜⎝

R∂e1∂φ1

. . . R ∂en

∂φ1

. . .

R∂e1

∂φn−1. . . R ∂en

∂φn−1

e1 . . . en

⎞⎟⎟⎟⎠⎛⎝

∂r∂l1. . .∂r∂ln

⎞⎠ = J (R, φ)

∂r

∂l⇒

∂r

∂l= J−1(R, φ)

⎛⎜⎜⎜⎝

∂r∂φ1

. . .∂r

∂φn−1∂r∂R

⎞⎟⎟⎟⎠ = J−1(R, φ)

(∂r∂φ

− r

R2

).

Consequently, as regards union sphere R = 1 equation (4) for r(l(φ), t) = r(φ, t) canbe rewritten as

∂+r(φ, t)

∂t= ρ

((− 1

r(φ,t)∂r(φ,t)

∂φ

1

)∣∣∣∣∣ J−1(1, φ)′F(t, r(φ, t)e(φ))

).

Example 3 (a bilinear system).Let

dx

dt= A(t)x(t) + B(t)u(t),

A(t) ∈ A(t), u(t) ∈ U(t), t ∈ [t0, t1], x(t0) ∈ X0,

where map A is continuous compact and convex valued map into Rn×n, B(t) ⊆ Rn×p isa given continuous matrix, X0 ∈ St (Rn), 0 ∈ int X0, map U is continuous compact andconvex valued map into Rp, and for any t ∈ [t0, t1] 0 ∈ int U(t).

S. Mazurenko

In general, since there is uncertainty in the matrix A(t), the reachability set is not convex.However, it can be shown that Property 1 is valid. Indeed, for any λ ∈ [0, 1]

λF(t, x) = λA(t)x(t) + λB(t)U(t) = A(t)(λx(t)) + B(t)(λU(t)) ⊆ F(t, λx).

Thus, for such directions l, in which r(l, t) is continuously differentiable it follows that

∂+r(l, t)

∂t= 1

r(l, t)ρ

(−∂r(l, t)

∂l

∣∣∣∣A(t)r(l, t)l + B(t)U(t))

)=

= maxA∈A(t)

⟨−∂r(l, t)

∂l, Al

⟩+ max

u∈U(t)

⟨−∂ ln r(l, t)

∂l, B(t)u

⟩.

Both these maxima are linear in the respective arguments. For example, consider polarcoordinates for n = 2, p = 1 :

J−1(1, φ) =(− sinφ cosφ

cosφ sinφ

).

If we define

A = {[a−i,j , a

+i,j ]}i,j=1,2, B =

(b1b2

)

then∂+r

∂t= r∑

i,j=1,2

maxai,j ∈[a−

i,j ,a+i,j ]

ai,j kiej + maxu∈[u−,u+]

ud,

whered = k1b1 + k2b2,

k1 = ∂ ln r

∂φsinφ + cosφ,

k2 = −∂ ln r

∂φcosφ + sinφ,

e1 = cosφ, e2 = sinφ.

In other words,

∂+r

∂t= r∑

i,j=1,2

(a−i,j + 1

2(sign(kiej ) + 1)(a+

i,j − a−i,j )

)kiej+

+(

u− + 1

2(sign(d) + 1)(u+ − u−)

)d,

where

sign(x) =⎧⎨⎩

1, if x > 00, if x = 0−1, if x < 0.

For an alternative approach to the reachability problem of bilinear systems in terms ofellipsoidal approximations please see [21].

5 Viability Problem

We will now proceed to the viability problem. Suppose there is observation Y (t) ∈ conv Rn

such that r(l|Y (t)) and ρ(l|Y (t)) are continuously differentiable functions of t . In this sec-tion we restrict our attention to trajectories that are viable, i.e. we would like to find thefollowing reachability set:

X [t] = {x[t] ∈ X[t] : ∀τ ∈ [t0, t] ⇒ x[τ ] ∈ Y (τ)}.

PDE for Evolution of Star-Shaped Reachability Domains

In other words, we are interested in only those trajectories of (2) that are consistentwith the observation Y over the whole period of time from t0 to t . We will assume thatX0 ⊆ Y (t0).

Theorem 4 [5] Suppose that for (2) Property 1 holds and there exists ry > 0 such thatryB0,1 ⊆ Y (t) for all t ∈ T . Then the reachability set X [t] of (2) is the only compact-setsolution to the following equation:

limσ→+0

σ−1h

⎛⎝X [t + σ ],

⋃x∈X [t]

{x + σF(t, x)}⋂

Y (t + σ)

⎞⎠ = 0, t ∈ T , X [t0] = X0.

(14)

Again, we would like to obtain a differential equation from (14) in terms of radialfunctions. If we now consider r(l, t) = r(l|X [t]) then the following theorem holds.

Theorem 5 Suppose that for (2) Property 1 holds and there exists ry > 0 such thatryB0,1 ⊆ Y (t) for all t ∈ T . Then for any non-zero direction l ∈ Rn, for which the radialfunctions r(l|Y (t)) and r(l, t) are continuously differentiable in l, there exits ∂+r(l, t)/∂t

that satisfies the following equation

∂+r(l, t)

∂t=⎧⎨⎩

1r(l,t)

ρ(

− ∂r(l,t)∂l

∣∣∣F(t, r(l, t)l))

, if r(l, t)<r(l|Y (t)),

min{

1r(l,t)

ρ(

− ∂r(l,t)∂l

∣∣∣F(t, r(l, t)l))

, ∂∂t

r(l|Y (t))}

, if r(l, t)=r(l|Y (t)).

(15)

Notice that the situation r(l, t) > r(l|Y (t)) is impossible since X [t] ⊆ Y (t). Thederivation of this equation will be carried out in the way similar to the previous theorem.

Proof Equation (14) is equivalent to the following two inclusions:

X [t + σ ] ⊆⋃

x∈X [t]{x + σF(t, x)}

⋂Y (t + σ) + σoσ (1)B0,1, (16)

⋃x∈X [t]

{x + σF(t, x)}⋂

Y (t + σ) ⊆ X [t + σ ] + σoσ (1)B0,1, (17)

where limσ→+0

oσ (1) = 0.

Consider (16):⋃

x∈X[t]{x + σF(t, x)}

⋂Y (t + σ) + σoσ B0,1 ⊆

⎛⎝ ⋃

x∈X[t]{x + σF(t, x)} + σoσ (1)B0,1

⎞⎠⋂(Y (t + σ) + σoσ (1)B0,1

).

Then the following inequality for radial functions should hold:

S. Mazurenko

r

⎛⎝ l|⋃

x∈X[t]{x + σF(t, x)}

⋂Y (t) + σoσ (1)B0,1

⎞⎠ ≤

≤ min

⎧⎨⎩r

⎛⎝ l|⋃

x∈X[t]{x + σF(t, x)} + σoσ (1)B0,1

⎞⎠ , r(l| Y (t + σ) + σoσ (1)B0,1

)⎫⎬⎭ .

However, we already obtained transformation of the first radial function in min in theprevious theorem. We will now apply Lemma 4 to obtain

r(l|Y (t + σ) + σoσ (1)B0,1) = r(l|Y (t + σ)) + σoσ (1).

Hence,r(l, t + σ) − r(l, t)

σ≤

≤min

{max

y∈Fε,z∈B0,1

(⟨∂r

∂l(x∗, t), l

⟩−1⟨∂r

∂l(x∗, t), y

⟩),r(l|Y (t + σ)) − r(l, t)

σ+oσ (1)

}+

+oσ (1).

Here we again used the fact that x∗ → r(l, t)l, consequently the multiplier of oσ (1) islimited.

1. If r(l|Y (t)) > r(l, t) then

limσ→+0

r(l|Y (t + σ)) − r(l, t)

σ= +∞

and the minimum will turn into the case of (9):

limσ→+0

r(l, t + σ) − r(l, t)

σ≤ 1

r(l, t)ρ

(−∂r(l, t)

∂l

∣∣∣∣F(t, r(l, t)l)

).

2. If r(l|Y (t)) = r(l, t) then we will have

limσ→+0

r(l, t + σ) − r(l, t)

σ≤ min

{maxy∈Fε

⟨− 1

r(l, t)

∂r

∂l(l, t), y

⟩,

∂

∂tr(l|Y (t))

}.

We again take infimum in ε > 0 and finally obtain:

limσ→+0

r(l, t + σ) − r(l, t)

σ≤ min

{max

y∈F(t,r(l,t)l)

⟨− 1

r(l, t)

∂r

∂l(l, t), y

⟩,

∂

∂tr(l|Y (t))

}.

Combining the two above cases we arrive at the following inequality:

∂+r(l, t)

∂t≤⎧⎨⎩

1r(l,t)

ρ(

− ∂r(l,t)∂l

∣∣∣F(t, r(l, t)l))

, if r(l, t) < r(l|Y (t)),

min{

1r(l,t)

ρ(

− ∂r(l,t)∂l

∣∣∣F(t, r(l, t)l))

, ∂∂t

r(l|Y (t))}

, if r(l, t) = r(l|Y (t)).

Consider (17). In terms of radial functions this inequality may be rewritten as

min

⎧⎨⎩r

⎛⎝l

∣∣∣∣∣∣⋃

x∈X[t]{x + σF(t, x)}

⎞⎠ , r(l|Y (t + σ))

⎫⎬⎭ ≤ r(l, t + σ) + σoσ (1).

PDE for Evolution of Star-Shaped Reachability Domains

However, according to (12)

r

⎛⎝l

∣∣∣∣∣∣⋃

x∈X[t]{x + σF(t, x)}

⎞⎠ ≥ r(l, t) +

⟨∂r

∂l(x−, t), l

⟩−1 ⟨∂r

∂l(x−, t), y(σ )

⟩σ + σoσ (1),

where limσ→+0

x−(σ ) = r(l, t)l and y(σ ) is a continuous function with y(0) = y0 ∈F(t, r(l, t)l). Consequently,

min

{⟨∂r

∂l(x−, t), l

⟩−1 ⟨∂r

∂l(x−, t), y(σ )

⟩+ oσ (1),

r(l|Y (t + σ)) − r(l, t)

σ

}≤

≤ r(l, t + σ) − r(l, t)

σ+ oσ (1).

1. If r(l|Y (t)) > r(l, t) then

limσ→+0

r(l|Y (t + σ)) − r(l, t)

σ= +∞

and the minimum will turn into:⟨∂r

∂l(r(l, t)l, t), l

⟩−1 ⟨∂r

∂l(r(l, t), t), y0

⟩≤ lim

σ→+0

r(l, t + σ) − r(l, t)

σ.

Again we will use homogeneity of r(l, t) and arbitrary choice of y0 ∈ F(r(l, t)l, t) toobtain

1

r(l, t)ρ

(−∂r(l, t)

∂l

∣∣∣∣F(t, r(l, t)l)

)≤ ∂+r(l, t)

∂t;

2. If r(l|Y (t)) = r(l, t) then we will have

min

{⟨∂r

∂l(r(l, t)l, t), l

⟩−1 ⟨∂r

∂l(r(l, t), t), y0

⟩,

∂

∂tr(l|Y (t))

}≤ ∂+r(l, t)

∂t.

We again take maximum over y0 ∈ F(r(l, t)l, t) and finally obtain:

min

{1

r(l, t)ρ

(−∂r(l, t)

∂l

∣∣∣∣F(t, r(l, t)l)

),

∂

∂tr(l|Y (t))

}≤ ∂+r(l, t)

∂t.

Combining the two above cases we arrive at the following inequality:

∂+r(l, t)

∂t≥⎧⎨⎩

1r(l,t)

ρ(

− ∂r(l,t)∂l

∣∣∣F(t, r(l, t)l))

, if r(l, t) < r(l|Y (t)),

min{

1r(l,t)

ρ(

− ∂r(l,t)∂l

∣∣∣F(t, r(l, t)l))

, ∂∂t

r(l|Y (t))}

, if r(l, t) = r(l|Y (t)).

which together with the opposite inequality concludes the proof.

6 Optimal Control Synthesis

Equation (4) can be used to solve optimal control problems. Consider the followingcontrollable system:

dx

dt= f (t, x, u), t ∈ [t0, t1]

S. Mazurenko

where x = (x1, . . . , xn−1), time interval [t0, t1] is fixed. The problem is to find optimalcontrol u∗ for the given initial and final sets X0, X1 while minimizing

t1∫t0

fn(τ, x, u)dτ → minu∈U

.

If we define

xn(t) =t∫

t0

fn(τ, x, u)dτ, x = (x, xn), X0 = X0 × {0},

then the optimization problem is equal to finding the minimal value of xn(t1) given thatx(t1) ∈ X1 and

dx

dt= f (t, x, u) = (f , fn)

′. (18)

If the exact reachability set X[t1] for (18) is known, we can intersect it with X1 ×R1 andthen take the minimal value of xn(t1). The respective point x∗

t1will be the end of the optimal

trajectory.Now in order to find optimal control that will result in x∗

t1one might use the following

result.

Lemma 5 Suppose that the conditions of Theorem 2 are satisfied for a system

dx

dt= F(t, x) =

⋃u∈U(t)

f (t, x, u), t ∈ [t0, t1], x(t0) ∈ X0.

Then for the following control

u∗(x, t) = arg maxu∈U(t)

⟨−∂r

∂l

∣∣∣∣l=x

, f (t, x, u)

⟩(19)

and respective trajectory x∗(t)dx∗

dt= f (t, x, u∗(x, t))

it follows that if ∃t ∈ [t0, t1] : x∗(t) ∈ ∂X[t] then ∀t ∈ [t0, t1] x∗(t) ∈ ∂X[t].Proof Indeed, let z(t) = r(x∗(t), t) and consider the following right-hand full derivative

dz

dt=⟨

∂r

∂l

∣∣∣∣l=x∗

,dx∗

dt

⟩+ ∂+r

∂t=

=⟨

∂r

∂l

∣∣∣∣l=x∗

, f (t, x∗, u∗)⟩+ 1

zmax

u∈U(t)

⟨−∂r

∂l

∣∣∣∣l=x∗

, f (t, x∗, u)

⟩=

=⟨

∂r

∂l

∣∣∣∣l=x∗

, f (t, x∗, u∗)⟩ (

1 − 1

z

)= g(t)

(1 − 1

z

).

Notice, that by definition z ≥ 1. The above ordinary differential equation has two typesof solutions: either z(t) ≡ 1, or

z − ln(z − 1) = C +t∫

t0

g(τ)dτ.

Since z(t) = 1 it follows that z(t) = r(x∗(t), t) ≡ 1, which is equivalent to x∗(t) ∈∂X[t].

PDE for Evolution of Star-Shaped Reachability Domains

Hence, if the conditions of Theorem 2 are satisfied (the origin in X0 can be approxi-mated by a ball εB0,1 with sufficiently small ε), the optimal trajectory of (18) satisfies thefollowing equation:

dx∗

dt= f (t, x∗, u∗(x∗, t)), x∗(t1) = x∗

t1, (20)

where u∗ is derived from (19). The overall procedure of control synthesis can be imple-mented backward in time with small steps �t :1. The terminal point x∗(τ ) is found as described above for τ = t1,

2. The optimal control u∗(x∗(τ ), τ ) is derived from (19),3. The previous point x∗(τ − �t) is restored from (20),4. Steps 2-3 are repeated for τnew = τold − �t until τold = t0.

7 Conclusion

In this paper we described the evolution of star-shaped reachability sets of differential inclu-sions in terms of the radial (Minkowski gauge) function. The derived equations are alsomodified to cope with measurement equations for available observations. Equations for therelated viability problem are then given. The solutions of the above equations can facilitatethe solution to the problem of optimal control synthesis.

The results presented above are true only for sets that contain the origin. This conditionseems to be technical since star-shaped sets can be defined for those with the center at anarbitrary point of Rn. The problem of finding the trajectory of such center of the star and itseffect on the results of this paper is still open.

Acknowledgments The author would like to thank A.B. Kurzhanski for his guidance and advicethroughout this research. This work was supported by the Czech Ministry of Education (LO1214).

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